Pr card type D

You can ask Commune.

You can ask at your commune though

No, it will not count. You'll have to wait 1 year and a half. (Knowing that in 6 months you'll have unlimited work permit)

It would have if you had a permanent resident permit in Italy, but you couldn't get that with 3 years of PhD (or if you had the European Blue Card but that's not your case as well).

If what you said is true, I can’t understand the logic behind this

The Logic is 7 yrs of residence as a student counts as 3.5 yrs for PR.

No, permanent resident permit in another Schengen State only remove the visa request to settle in another state. That's basically all. You won't get permanent resident permit until you respect the national law on that topic in your new country, you'll start from 0 (temporary resident - 1 year card - blablabla).

There's one useful additional info, is for the D card owner (now the "L" card) (permanent resident that have worked for 5 years+ straight & higher salary), those one will have way less administration than all other resident permit. They're nearly like the EU moving in another Schengen state.

And depending on the country, you'll lose your permanent resident status (most likely after 2 years out of the country)